Prove that if $a^2 + ab + b^2 = 0$ then $a = b = 0$?

Question

We are given that $a, b \in F_{2^n}$ where $n$ is an odd +ve integer. Suppose $a^2 + ab + b^2 = 0$ then we have either $a = 2^n-b^2$ or $a+b = 2^n - b^2$. Which implies that $\sqrt{2^n -a} = +-b $ or $b = 2^n - a$. Thus $b = 0$ and $a= 2^n= 0$

Is it correct? Is there a simple proof?